Valley Hall effect in a bilayer graphene phononic crystal without localized Berry curvature

Valley degree of freedom, as the energy extreme point of $K$ and ${K}^{\ensuremath{'}}$ momenta that are distinctly separated in reciprocal space, gives rise to negligible valley scattering, and thus the robust valley edge states occur at the interface between two domains with opposite valley Chern numbers. The valley Chern numbers are the integration of nonzero Berry curvature around $K$ and ${K}^{\ensuremath{'}}$. In this work we report an acoustic scheme of achieving the valley Hall effect without localized Berry curvature at $K$ and ${K}^{\ensuremath{'}}$ in a bilayer graphene phononic crystal. Since the valley Chern number is zero, nontrivial valley Euler numbers are employed to characterize the topology of the phononic crystals. We finally demonstrate that the pseudospin and edge state arise from the topological phase transition of the nonzero valley Euler numbers. Our work provides a macroscopic platform to explore the intriguing properties of the valley Hall effect.